Hydraulic differentiation



y 1961 R. OLDENBURGER ET AL 2,992,650

HYDRAULIC DIFFERENTIATION Filed March 28, 1960 2 Sheets-Sheet 2 lA/ ur 5/04 44 (230? VflR/HBAE SPEED I I I I IN V EN TORS."

2,992,650 HYDRAULIC DIFFERENTIATION Rufus Oldenburger, Lafayette, Ind., and George Forrest Drake, Rockford, 111., assignors to Woodward Governor Company, Rockford, 111., a corporation of Illinois Filed Mar. 28, 1960, Ser. No. 17,835 16 Claims. (Cl. 137-85) This invention relates in general to hydraulic systems, and is more particularly concerned with the creation of a hydraulic signal which varies as the time derivative of an input signal or changeable physical condition.

In the art of servo control systems it is often desirable to effect a corrective or self-balancing action according not only to the magnitude of the departure or error of the controlled condition, but also according to the first or higher order time derivatives (rate of change) of the error. As is well known, this produces a compensating or anticipating action which reduces hunting about the balance point. Moreover, in many instrumentation systems it is desired to display or record the time derivative or rate of change of a variable quantity or condition.

There are known in the art several different mechanical and electrical devices which produce time derivative signals. In the field of hydraulic controls and instruments, however, the prior differentiation arrangements, such as compensating dashpots, have been burdened with serious disadvantages, for example, the introduction of lags which to a large measure cancel or offset the derivative or lead effect.

It is the general aim of this invention to bring forth a new and improved method and apparatus for producing a hydraulic signal which varies as the time derivative of a changeable input signal or condition.

Coordinate with that aim, it is an object of the invention to provide differentiation of hydraulic signals while making possible the convenient changing of the relative strength of the derivative signal without afiecting the other lags or leads of a control system.

Another object of the invention is to achieve hydraulic differentiation which is reliable and relatively precise, and which may be effected by a wide variety of simple and economical physical systems.

It is still another object of the invention to provide a method and apparatus for producing a composite output signal (e.g., a variable pressure) which varies as the algebraic sum of the changes in a variable condition and the time derivative of such changes.

Other objects and advantages will become apparent as the following description proceeds, taken in conjunction with the accompanying drawings, inwhich:

FIGURE 1 is a diagrammatic illustration of a hydraulic differentiation system embodying the features of the invention;

FIG. 2 is a graph depicting the pressure vs. the flow characteristic of an orifice employed in the system of FIG. 1;

FIG. 3 is a diagrammatic illustration of an alternative embodiment of the invention, and one which provides not only a derivative signal but a signal which varies as the algebraic sum of the variable condition and its time derivative; and

FIG. 4 is a simplified diagram corresponding to a portion of FIG. 1.

While the invention has been shown and will be described in some detail with reference to particular embodiments thereof, there is no intention that it thus be limited to such detail. On the contrary, it is intended here to cover all modifications, alternatives, and equivalents falling within the spirit and scope of the invention as defined by the appended claims.

The method and apparatus for producing a hydraulic nited States Patent pressure which varies as the time derivative of a changeable condition or input signal may be understood with reference to the simplified embodiment shown schematically in FIG. 1. In the differentiating procedure, a hydraulic pressure is first created which varies according to the value of a changeable input signal or physical condition. It is assumed in connection with FIG. 1 that the angular position of a control member or knob 10 is the variable physical condition, the angle on of the knob representing the deviation of the condition from a reference value or condition.

To create a fluid pressure p which varies (from an initial or reference value) according to the angle or of the knob 10, the latter is associated with and forms the adjustment member for a pressure regulator 11. This pressure regulator may take any of a variety of forms known to those skilled in the art, and its details thus need not be shown or described. Briefly stated, the regulator 11 is a position-to-pressure transducer. It has an input conduit 12 connected to a source of pressure fluid here shown as a pump 14, and a venting conduit 15 leading back to a fluid sump 16. The pressure p in an output line or conduit 18 of the regulator 11 is proportional to the angular position a of the adjustment member or knob 10. The pressure regulator 11, therefore, forms a transducer which creates an output pressure p which varies directly according to a changeable physical condition which in the present instance is the angular position a of the control knob 10.

Next, the volume of a body of liquid is changed in accordance with changes in the pressure p For this purpose, the output conduit 18 of the pressure regulator 11 is connected to communicate with the interior of a chamber 19 formed with a resiliently deformable or movable wall. As here shown, the chamber 19 is closed at one end by a resilient, deformable bellows 20 which is backed up and urged to the left by a compression spring 21. As the pressure p increases or decreases, therefore, the bellows 20 will be correspondingly contracted or expanded, and the volume of fluid in the chamber 19 will be correspondingly increased or decreased. When the end wall of the bellows 20 is stationary, the pressure p in. the chamber 19 equals the pressure p produced by the regulator 11. The spring 21 assures that the pressure p is always equal to the pressure p except for a negligible time lag.

Further in accordance with the invention, the changes in the volume of the fluid in the chamber 19 are consummated by a fluid flow through a restriction or orifice 22 which has a linear pressure vs. flow characteristic. While the orifice 22 may take a variety of forms, it is here shown as a cylindrical, pipe-like orifice of relatively small diameter compared to its length. Such an orifice has a pressure vs. flow characteristic illustrated in FIG. 2. As shown by the line 24 in FIG. 2, if the flow Q (volume per unit time) through the orifice 22 increases from zero in either direction, then the pressure drop p across the orifice correspondingly increases in proportion and in the same sense. The line 24 represents a proportional or linear variation of pressure p;; with the flow Q, and may be mathematically defined by the equation p =cQ, Where 11 is the pressure drop across the orifice 22, Q is the flow therethrough and c is a constant of proportionality.

As an exemplary device which utilizes the hydraulic derivative signal, an indicator or pressure gauge 25 is shown in FIG. 1 having conduits connected to the opposite sides of the orifice 22. The pressure gauge 25 thus indicates or displays the sense and magnitude of the pressure drop 2 across the orifice.

In operation, as the angle a of the knob 10 is varied, the pressure 2, will correspondingly vary and fluid will flow into or out of the chamber 19 until the pressure p and pressure p are equal. When the pressure p changes, the bellows 20 will expand or contract a corresponding amount, and the volume of fluid in the chamber 19 will be decreased or increased proportionally. The flow of fluid through the orifice 22 occasioned by this change of volume, i.e., by contraction or expansion of the bellows 20, will create a pressure drop across the orifice 22 which is proportional to the flow. The magnitude of the flow Q depends upon the rate at which the bellows 20 expands or contracts, and this in turn depends directly (except for a negligible time lag) upon the rate of change of the pressure 12 Accordingly, the flow Q through the orifice 22 and the pressure drop p thereacross will vary substantially as the time derivative of the pressure p and since the latter is proportional to the variable condition or angle or, the pressure drop p will vary in sense and magnitude as the first time derivative,

du dt of the angle oz.

The differentiation method and apparatus illustrated by FIG. 1 will not respond with mathematical precision to a step change in the angle a, since that would involve a time derivative which in value approaches infinity. However, no physical ditferentiation system, whether electrical, mechanical or hydraulic is theoretically perfect in its response to instantaneous step changes of the input, and the present system is sufiiciently precise for practical control or instrumentation devices. Moreover, as a practical matter, it is impossible to vary many conditions, such as the angle a of the knob 10, with an instantaneous step change because finite physical or electrical inertias are involved. For practical purposes, therefore, the pressure drop 1 across the orifice 22 will vary substantially as the rate of change of the angle a. The magnitude and sense of the time derivative,

will thus be indicated by the utilization device or gauge 25.

In the foregoing discussion it has been assumed that the several conduits connecting the pressure regulator 11, the chamber 19 and the gauge 25 with the orifice 22 are relatively large in diameter compared to the orifice 22 and offer only a negligible resistance to the flow of fluid. Moreover, it is assumed that the gauge 25 operates to display the magnitude of the pressure p without requiring an appreciable flow of fluid through the conduits which connect it to the orifice 22.

While the method and apparatus depicted by FIG. 1 will be clear from the foregoing, a brief mathematical analysis will be helpful to a better understanding of the diflerentiating operation. First, it will be clear that as the different pressures vary, the pressure 1 will always be equal to the sum of the pressures p and p This may be written:

The amount the spring 21 is compressed, i.e., the position of the movable end Wall of the bellows, depends upon the pressure p in the chamber 19 and upon the modulus or constant k, of the spring. In other words, the sum of the forces on the end wall of the bellows (the fluid pressure force and the spring force respectively, urging the end Wall to the right and left) must be zero, and the end wall will always shift until such balance is obtained. This may be expressed:

where p =pressure in the chamber 19 a=area of bellows end wall k =modulus of spring 21 x=position of bellows end wall measured from a reference position.

Further, the pressure drop 12;, across the orifice 22 is proportional to the flow Q therethrough, as previously noted:

Where c is a constant of proportionality. The flow Q (volume per unit time) depends, however, on the rate of change of volume of the chamber 19, and thus upon the rate at which the bellows end wall changes its position. Expressed mathematically, this becomes Q=a ax This may be rewritten by taking Q from Equation 3:

1 c Q=- Pa Substituting x from Equation 5 into Equation 2:

And substituting p from Equation 6 into Equation 1:

Pa=-* Pi k c D+ 1) where D is a differential operator,

From Equation 7.3 it will be apparent that the pressure drop 2 is proportional to the time derivative of the pressure 17 i.e., p The expression in parentheses in Equation 7.3 represents a time lag, and indicates that the output pressure signal 17 varies as the derivative p but with some delay. This time delay, however, is not a factor which necessarily obviates the usefulness of the derivative variations in the pressure 2 particularly where the rate of change of the signal p is not expected to be especially great. Thus, the analysis given above demonstrates that the simple method and apparatus of FIG. 1 serves to produce a hydraulic signal (variable pressure p which varies as a time derivative of an input variation, i.e., the angle on or the pressure p I Referring now to FIG. 3, an alternative embodiment of the invention is there disclosed which illustrates a more precise and preferred method and apparatus for producing a hydraulic pressure which varies as the time derivative of a changeable signal or condition. Moreover, the procedure and arrangement illustrated by FIG. 3 makes possible the creation of a hydraulic pressure which varies as the algebraic sum of the condition and its time derivative. The changeable physical condition in the arrangement of FIG. 3 is the speed of a shaft 30 driven from some means (not shown) such as a variable speed engine or motor.

As a first step in the diiferentiating procedure, a liquid pressure is created which varies in proportion to changes in the speed of the shaft 30. For this purpose a selfbalancing speedtopress'ure transducer is employed.

While such transducer may take a variety of forms, it is here shown as including a cylindrically shaped, rotatable body 31 driven from the shaft 30 through gears 32, 33. The body 31 rotates in a stationary housing 34 and is formed to define a valve chamber 35 having an output port 36, and a balancing chamber 38. Movably disposed in the body 31 is a plunger 39 formed with a valve element 40 and a balancing piston 41. The balancing chamber 38 and the valve chamber 35 are separated by a fluidsealing land 42 which is formed as an integral part of the plunger 39.

With the plunger 39 in a neutral position, the valve element 40 covers and closes the output port 36. If the plunger is shifted downwardly from the neutral position, however, the element 46 partially uncovers the output port 36 and connects the latter to a conduit 44 containing pressurized fluid produced by a suitable fluid pressure source such as a pump 45. The pump has its intake communicating with fluid in a sump 46, and its output is returned to the sump through a relief valve 48 so that the pressure in the line 44 is maintained substantially constant. When the valve member 40 is shifted upwardly from the neutral position, the output port 36 is placed in communication with a return conduit 49 which permits fluid to flow from the port "36 back to the sump 46.

The plunger '39 is normally urged downwardly by a spring S1. This downward force of the spring is counterbalanced by the upward force exerted on the under side of the balancing piston 41 by the pressure of fluid in the balancing chamber 38. Moreover, an upward force is exerted on the plunger 39 by flyweights 50 pivoted at 51 to the body 31 and having inwardly extending arms 52 which bear against the underside of a disk 54 integral with the plunger. The centrifugal force urging the flyweights 50 outwardly, and upward force exerted by the arms 52 on the disk 54, vary according to the rotational speed of the body 31 and the shaft 30.

The output port 36 of the valve chamber 35 is connected through the housing 34 by an annular cavity 53 Which communicates via fluid conduits 56, 57 and 58 with the balancing chamber 38. The stationary conduit 58 leads through the housing 34 to an annular cavity 59 and a passage 60 so that the conduit 58 and the balancing chamber 38 are always in direct communication even though the body 31 is rotating. Thus, as the speed of the shaft 30 and the body '31 increases or decreases so that the upward force exerted on the plunger by the flyweigh-ts 50 increases or decreases, the plunger 39 will tend to be shifted upwardly or downwardly. This will respectively connect the port 36 either to the return conduit 49 or to the pressure fluid output of the pump 45, permitting fluid to be vented from or introduced to a balancing chamber 38 through the conduits 56, 57 and 58. The pressure within the balancing chamber will thus be respectively decreased or increased until the plunger 39 is returned to its neutral position. This self-balancing action occurs relatively fast, and because the movement of the plunger 39 is over a relatively small distance, the flow of fluid out of or into the balancing chamber 38 does not involve large quantities of fluid. As a practical matter, therefore, the pressure within the balancing chamber 38 varies (except for a short time lag) propon-- tionally with changes in the speed of the shaft 36 as sensed by the flyweights 50.

The upper end of the spring S1 bears against an adjustable abutment 61 selectively positioned in a vertical direction by a rack 62 and a pinion 63, the latter being connected to a manually adjustable lever 64. By adjusting the lever 64 and changing the position of the abutment 61, the force exerted by the spring S1 downwardly on the plunger 39 and the value of the fluid pressure in the balancing chamber 38 for a particular speed of the shaft 30 can be adjusted. The conduit 58 as here shown includes a restriction 58a which ofiers appreciable impedance to the flow of fluid only at very high frequency pressure variations. This is a damping restriction which for practical purposes is effective only at frequencies much greater than the expected frequency of variations in the speed of the shaft 30, and which serves to prevent instability or oscillation of the plunger 39.

Summarized, the speed-to-pressure transducer thus far described in connection with FIG. 3 produces a fluid pressure in the balancing chamber 38 which is proportional to the speed of the input shaft 30. This pressure is created by the self-balancing action of the valve element 46 which increases or decreases the pressure in the balancing chamber 38 to restore the plunger 39 to a neutral position whenever the speed of the shaft 30, and the force on the plunger created by the flyweights 50, increases or decreases.

As a second step in the differentiating procedure, the volume of a body of liquid is subjected to the pressure in the balancing chamber 38 which, as described above, varies according to the speed of the shaft 30. This is accomplished in the exemplary arrangement of FIG. 3 by a chamber 65 connected between the conduits 57 and 58 and containing a body of liquid which is thus subjected to the same pressure as that which exists in the balancing chamber 38.

In order to change the volume of the chamber 65 according to the pressure therein, the chamber 65 is formed with a resilient yieldable wall portion, and for this purpose is here constructed to have one wall 66 (with an area a carried at the end of a deformable bellows 67. The bellows contains a spring S2 (having a spring constant k which urges the wall 66 downwardly, but permits the wall 66 to shift upwardly as the pressure in the chamber 65 increases. As the pressure in the balancing chamber 38 and the second chamber 65 increases or decreases, the bellows 67 contracts or expands a proportional amount, thus correspondingly changing the volume of fluid in the chamber 65.

In accordance with the present differentiating procedure, such changes of volume in the chamber 65 are consummated by causing fluid flow through a restriction or orifice having a linear pressure vs. flow characteristic. For this purpose an orifice 69 (similar to the orifice 22 previously described) is interposed between the conduits 56 and 57. As the pressure in the chamber 38 is increased or decreased in proportion to increases or decreases in the speed of the shaft 30, the pressure in the chamber 65 is correspondingly increased or decreased so that the bellows 67 contracts or expands. This increases or decreases the volume of fluid in the chamber 65, and such volume changes are consummated by the fluid flow through the orifice 69. Such flow comes from the pump 45 and fluid pressure line 44 when the pressure in the chamber 65 is increased, or passes through the valve port 36 and the conduit 49 to the sump 46 when the pressure in the chamber is decreased. This fluid flow through the orifice 69 creates a pressure drop across the latter which varies substantially as the time derivative of the speed of the shaft 30. The magnitude and sense of the time derivative signal may be indicated by a pressure gauge 70 which is similar to the gauge 25 shown in FIG. 1.

Whenever the speed of the shaft 30 changes from one value to another, so that the plunger 39 is momentarily shifted from its neutral position to open the valve 40 in one direction or the other, the fluid which flows through the valve port 36 (either from the pump 45 or into the sump conduit 49) must be sufiicient in total volume to match the change in volume of the chamber 65. The pressure in the balancing chamber 38 will thus not be changed to an equilibrium value instantaneously, but will require some finite although very short and negligible time delay before the valve plunger 39 is restored to its neutral position. For practical purposes, therefore, the pressure in the chamber 65 and the balancing chamber 38 will vary'as the speed of the shaft 30. Moreover, there will be some fluid flow into or out of the balancing chamber 38 as the plunger moves upwardly or downwardly, such flow passing through the orifice 69. However, the eflective area al of the balancing piston 41 (that is, the area of the underside of the piston 41 less the area of the upper side of the land 42) is made so small relative to the area 11 of the bellows end wall 66 that the flow through the orifice 69 is due, for practical purposes, entirely to changes in the volume of the chamber 65. In other words, flow through th orifice 69 due to changes in the volume of the balancing chamber 38 is so small that it can be neglected, and the pressure drop across the orifice 69 is thus proportional to the much larger flow due to changes in the volume of the chamber 65.

This relationship is illustrated in the simplified diagram of FIG. 4. With the effective area a of the balancing piston 41 (FIG. 3) much smaller, say ten or fifteen times smaller, than the area a of the bellows wall 66, it may be assumed that there is no appreciable fluid flow through the conduit 58. Rather, the pressure p in the conduit 58 varies (except for a small time lag) in proportion to the speed of the shaft 30 (FIG. 3). This, as the shaft speed and pressure p change, the pressures p and p in the conduit 58 and chamber 65 correspondingly change, and the bellows 67 correspondingly expands or contracts. This change in the volume of the chamber 65 produces fluid flow to or from the sump 46 through the conduits 5-6, 57 and the orifice 69 connected therebetween. Because the flow Q is proportional to the rate of movement of the bellows wall 66 which in turn is proportional to the rate of change of the pressure p the pressure 1 across the orifice 69 varies substantially proportionally to the time derivative of the pressure p and the time derivative of the speed of the shaft 30. The spring S2 as shown in FIG. 4 is welded at its opposite ends to the bellows and the chamber so that it can act in tension or compression as the pressure 11 varies above or below atmospheric pressure.

With this explanation of FIG. 4, it will be understood that the gauge 70 in FIG. 3 displays the value of the pressure drop 1);. and thus gives an indication of the magnitude and sense of the time derivative of the variable condition, i.e., the rate of change of the speed of the shaft 30.

In order to create a fluid pressure variation and a net output signal which changes as the algebraic sum of the speed of the shaft 30 and its time derivative, the pressure in the chamber 65 is added to the pressure drop across the orifice 69. This is accomplished in the exemplary system of FIG. 3 by connecting a sensing conduit 72 to communicate with the left side of the orifice 69, i.e., that side which is remote from the chamber 65. The conduit 72 leads to the interior of an output chamber 74 which contains a resilient, compressible bellows 75 backed up by a compression spring S4 (having a spring constant k The bellows is rigidly connected to a final output rod 76. With this arrangement, the pressure in the sensing conduit 72 and the output bellows chamber 74 varies as the algebraic sum of the pressure p in the chamber 65 and the pressure drop p across the orifice '69. The modulus k or stiffness of the spring S4 is chosen to be relatively great so that the bellows 75 does not expand or contract over an appreciable distance and the fluid flow into or out of the conduit 72 is negligible in comparison to the fluid flow through the orifice 69. Because the bellows 74 expands and contracts in proportion to the pressure supplied to the chamber 74 through the conduit 72, the physical position of the output rod 76 changes in proportion to the algebraic sum of the pressure in the chamber 65 and the pressure drop across the orifice 69. Thus the position of the rod 76 changes in proportion to the algebraic sum of the speed and the time derivative of the speed of the shaft 30.

While the organization and operation of the system shown in FIG. 3 will be apparent from the foregoing description, a brief mathematical analysis may be helpful in reaching a better understanding of the operation.

Consider first the summation of vertical forces on the plunger 39 in FIG. 3. The force equation may be written:

where:

k the combined scale of the spring S1, the flyweights 50 and the hydraulic reaction of the valve 40;

x=the displacement (positive up) of the plunger 39 from :a neutral position in which the element 40 closes the port 36;

p =the pressure change from a reference value within the chamber 38 and beneath the land 41;

a =the effective area on the underside of the land 41;

k =the proportionality constant relating speed of the shaft 30 and force exerted on the plunger 39 by the flyweights 50;

=the deviation of the speed of the shaft 30 from a selected reference value.

Since, as previously indicated, the restriction 58a may be neglected, the pressure p in the balancing chamber 38 may be validly assumed to be always equal to the pressure p in the chamber 65. Thus.

Equating the vertical forces on the bellows wall 66 (an upward force by fluid pressure and a downward force by the spring S2), this may be written:

where k is the combined modulus of the spring S2 and the bellows 67; and y is the displacement (positive down) of the bellows wall measured from a reference position.

As the valve element 40 is opened relative to the port 36, the pressure drop through the opening may be assumed to be substantially constant. The flow Q through the port 36 thus depends directly upon how wide the valve port is opened, and the direction (positive to the right) of flow depends on whether the plunger displacement x is positive or negative. The flow Q through the port 36 may thus be expressed:

the movable member or wall therein. Thus,

d1: d dZ Q 1Ei 2E% 4g? which may be rewritten with the operator D representing derivatives where an; equals the area of the movable wall of the bellows and Z equals the displacement of the wall from a reference position. As previously explained, the area 122 is made relatively small compared to the area a so that the first term on the right side of Equation 12.1 is negligible in comparison to the second term. More over, the spring constant k; for the spring S4 is made so large that the variable Z and its derivatives are small. Therefore, the flow into or out of the chambers 38 and 74 is quite small and indeed negligible in comparison to the much larger flow into or out of the chamber 65;

Thus, Equation 12.1 may be rewritten by neglecting the first and third terms on the right hand side:

The foregoing equation indicates, on the basis of the stated assumptions, that all significant flow Q passes through the orifice 69 into or out of the chamber 65. That orifice has the linear characteristic:

Pa= Q Where p is the pressure drop across the orifice and c is a constant of proportionality determined by the diameter and length of the orifice. Replacing Q from (12.2) in Dym (14.1)

y MD The expression for y from (15) is substituted in Equation to arrive at:

PI= T DP3 And by equating the flow Q as expressed in Equations 11 and 12.2 we obtain:

Now, substituting x from (18) and p from (16) into Equation 8, we write:

which reduces to:

n- M M 12 (1 2) lo e (1 0 70 02 lc a lc 1 This latter expression has the form:

where K is a composite constant and where T is a composite time constant From Equation 20 it will be apparent that the pressure drop p across the orifice 69, and the reading of the gauge 70, vary as the time derivative D of the input variable i.e., the speed of the shaft 30. There is a time lag in this variation which is represented by the time constant T but the lag can be made acceptably small for practical purposes. This is done by making T small, i.e., making k large and making k small. Such adjustments can be made Without affecting the gain factor K, and the gain may be made relatively great by making k and c relatively large. Thus, the time lag T and the gain K or derivative signal strength may be individually adjusted to acceptable practical values. In other words, by making the coeflicient k of the valve 40 large and by making the combined scale factor k small, the time lag of the differentiating system may be reduced to a low value. Adjustments in the constants a a and k may also be made to determine the magnitude of the time lag T although these will cause corresponding changes in the gain factor K. Summarized, a very precise differentiation Which responds faithfully to relatively rapid changes of the input condition is thus realized. Of course, any utilization device, besides the gauge 70 in FIG. 3, may be connected to receive the derivative pressure variation across the orifice 69.

To understand how the pressure 17 and the displacement Z of the rod 76 vary as the algebraic sum of the changes and rate of change of the speed of the shaft 30, it may be recognized first that the pressure p; is always equal to the sum of the pressure p in the chamber 65 and the pressure drop p That is:

By combining Equations 12.2 and 13, 12 may be expressed:

ps= z y (24) Substituting p from (24) and p from (10) into Equation 23:

Now substituting p from (26) and x from (27.1) into Equation 8:

The expression for y in (28.2) may be substituted into Equation 25.1 to derive:

2 E i 3+ 1) k 10 p1=;;-,-,- (2

k k a Equation 29 has the general form:

K D 1 p.=K1( T: 0)

Where K is a composite gain constant kg 1= a and where K and T are composite constants It will thus be apparent from Equation 30 that the pressure 17 and the displacement Z of the bellows 75 and rod 76 in FIG. 3 vary proportionally with the algebraic sum of the speed changes and rate of speed changes of the shaft 30. That is, the numerator of Equation 30 shows that p, varies as the sum The denominator of Equation 30 shows, however, that this variation of pressure 17 is affected by a time lag, with a time constant T This is the same as the time constant T shown by Equation 22, and it can be reduced to a low value in the manner previously described. Thus, the method and apparatus depicted by FIG. 3 provides a final signal (p; or Z) which varies as the algebraic sum of a variable (speed of shaft 30) and the time derivative of that variable, the time delay involved being reduced to a practically insignificant size. The procedure and equipment for producing this result are, nevertheless, simple and reliable in their operation.

Moreover, the relative strengths of the proportional signal (K 'qb) and the derivative signal (K K D) in contributing to the final output signal (p; or Z) may be readily adjusted. The relative magnitude or strength of the proportional component (Km) may be selected by adjusting the ratio of the constants k and al as indicated by Equation 31. Then, by increasing or decreasing the composite constant K e.g., by increasing or decreasing the orifice coefiicient c, the derivative signal may be made stronger or weaker relative to the speed-proportional signal. For laminar flow through a pipe-like orifice, the constant of proportionality relating the pressure drop and flow, i.e., c, is directly related to the length of the orifice and inversely related to the fourth power of the orifice diameter. Thus by changing either the length or the diameter, or both, of the orifice 69, the constant c in Equation 32 may be increased or decreased, and thus the constant K in- Equation 30 given the desired value.

The movable rod 76 which changes in position according to the algebraic sum of the speed deviation and first time derivative of speed deviation of the shaft 30 may be utilized in a servo control system which functions to keep the speed of an engine or motor driving the shaft 30 substantially constant. Such a servo control system or speed governor is more fully disclosed and claimed in applicants copending application Serial No. 609,944, filed September 14, 1956, now Patent 2,931,342, issued April 5, 1960, the present application being a continuation-inpart of such copending application. Because the organization and details of the complete servo control system are not a part of the invention here disclosed and claimed, those details have not been illustrated and need not be described. It is sufiicient only for purposes of the present application to understand that in many hydraulic control or instrumentation systems there is need to produce a hydraulic pressure variation which changes as the first derivative of an input signal or variable condition. That is reliably and simply accomplished by the method and apparatus here disclosed since the pressure drop across the orifices 22 or 69 (in FIGS. 1 and 3) will vary substantially as 'the derivative of the changeable condition (the position of the knob in FIG. 1, or the speed of the shaft 30 in FIG. 3). Moreover, in utilizing a chamber containing a body of fluid and changing the 1'2 pressure in that chamber so as to correspondingly change its volume and create fluid flow through an associated linear characteristic orifice, the present invention makes possible the production of a composite output signal such as the pressure 7 in the chamber 74. Such pressure and the displacement or the rod 76 vary as the algebraic sum of the variable condition and the time derivative of the variable condition.

We claim as our invention:

1. The method of creating a hydraulic pressure variation substantially proportional to the first time derivative of a variable condition, comprising changing the volume of a body of fluid at a rate substantially proportional to the first time derivative of variations of said condition, and consummating such volume changes by causing fluid to flow into or out of said body through an orifice having the characteristic p=cQ where p is the pressure drop across the orifice, Q is the rate of fluid flow therethrough, and c is a constant, the pressure drop across the orifice thus being proportional to the first time derivative of said condition.

2. The method of creating a hydraulic pressure variation substantially proportional to the first time derivative of a variable condition, comprising changing the pressure of a body of fluid from a reference pressure value in proportion to changes in the value of the variable condition, changing the volume of the fluid body in proportion to the pressure variations therein, and consummating such volume changes by causing fluid to flow into or out of the fluid body through a pipe-like orifice, whereby the pressure drop across the latter varies as the first time derivative of changes in the variable condition.

3. Apparatus for producing a hydraulic pressure variation proportional to the time derivative of a variable condition, comprising, in combination, a fluid chamber, means for changing the volume of said chamber in proportion to changes in said condition, and an orifice communicating with said chamber and through which fluid flows as the volume of the chamber changes, said orifice having the characteristic p=cQ where p is the pressure drop across the orifice, Q is the rate of flow through the orifice, and c is a constant, whereby the pressure drop across the orifice varies in proportion to the time derivative of changes in said condition.

4. Apparatus for producing a hydraulic variation proportional to the first time derivative or a variable condition, the combination comprising a fluid chamber, means for varying the pressure of fluid within said chamber in proportion to changes in said condition, a resilient, compressible wall within said chamber which is expanded or contracted in proportion to the variations of pressure in said chamber to proportionally change the volume of the latter, and a pipe-like orifice communicating with said chamber and through which fluid flows as the volume of the chamber changes, whereby the pressure drop across said orifice varies in proportionto the time derivative of changes in said condition.

5. The method of creating a hydraulic pressure variation substantially proportional to the first time derivative of a variable condition, comprising changing the pressure of liquid in a chamber according to variations of said condition, changing the volume of the chamber according to changes of the pressure therein, and passing fluid flow resulting from changes in said volume through an orifice having a linear pressure vs. flow characteristics, so that the pressure drop across said orifice varies as the time derivative of changes in said condition.

6. A hydraulic diiferentiator for producing a pressure variation substantially proportional to the time derivative of the changes in a variable condition, comprising in combination, a fluid chamber, means for varying the pressure in said chamber in accordance with changes in said condition, means associated with said chamber for changing its volume in accordance with changes in the pressure therein, and an 'orifice having a linear pressure 13 vs. flow characteristic communicating with said chamber and through which fluid flows as a result of changes in said chamber volume, whereby the pressure drop across said orifice varies as the first time derivative of changes in said condition.

7. The method of creating a hydraulic pressure variation proportional to the first time derivative of a variable condition comprising converting changes in said condition into corresponding changes in a liquid pressure, applying said changes in pressure to body of liquid, changing the volume of the body of liquid according to changes of pressure therein, and effecting such changes in the volume of the liquid body by flow into or out of the same through an orifice having a linear pressure vs. flow characteristic, whereby the pressure drop across said orifice varies as the derivative of changes in said condition.

8. Apparatus for creating a hydraulic pressure which varies as the time derivative of changes in a variable condition, comprising, in combination, a self-balancing transducer for converting changes in said condition into corresponding changes in liquid pressure within a chamber, resilient means associated with said chamber for changing the volume of the latter in accordance with changes of pressure therein, and an orifice having a linear flow vs. pressure characteristic communicating with said chamber and through which fluid flows to or from said transducer as the volume of said chamber changes, whereby the pressure drop across said orifice varies as the derivative of changes in said condition.

9. A hydraulic diflierentiator comprising, in combination, a self-balancing transducer for converting changes of a variable condition into corresponding changes in a fluid pressure, said transducer having a plunger forming a valve element and a balancing piston respectively movable in a valve chamber and a balancing chamber, said valve chamber being adapted for connection to a source of fluid pressure and a sump and having a port respectively connected to the latter as said plunger is shifted in opposite directions from a neutral position, means for applying a force to said plunger which changes in accordance with changes in said condition from a reference value, an orifice having a linear flow vs. pressure characteristic connected between said port and said balancing chamber, a second chamber connected through said orifice to said port, resilient means associated with said second chamber for making the volume of the latter change in accordance with the pressure therein, said balancing chamber and piston being sized relative to said second chamber such that the fluid flow into or out of the balancing chamber in response to a change in said condition is negligible relative to the resulting fluid flow into or out of said second chamber, whereby the pressure drop across said orifice varies substantially as the derivative of changes in said condition.

10. The method of creating a hydraulic pressure variation substantially proportional to the sum of the changes and the first time derivative of the changes of a variable condition, comprising varying the pressure in a body of fluid in proportion to the changes of the variable condition, changing the volume of the fluid body in proportion to the pressure variations therein, consummating such volume changes by causing fluid to flow into or out of the chamber through an orifice having a linear pressure vs. flow characteristic, and creating a net fluid pressure which varies as the algebraic sum of the pressure in said fluid body and the pressure drop across said orifice, whereby said net fluid pressure varies as the algebraic sum of the changes and the first derivative of the changes of said condition.

11. The method of creating a hydraulic pressure variation proportional to the.sum of the changes and the first time derivative of the changes in a variable condition, comprising changing the pressure of liquid in a chamber according to variations of said condition, changing the Volume of the chamber according to changes of the pressure therein, passing fluid flow resulting from changes in said volume through an orifice having a linear pressure vs. flow characteristic, and producing a not pressure which varies as the algebraic sum of the pressure in said chamber and the pressure drop across said orifice.

12. The method of creating a hydraulic pressure variation substantially proportional to the algebraic sum of the changes and the first time derivative of changes in a variable condition, comprising varying the pressure in a fluid body from a reference pressure value in proportion to changes of the variable condition, confining said fluid body in a chamber having a resilient, deformable wall so that its volume changes in proportion to the pressure therein, consummating such volume changes by causing fluid to flow into or out of the body through a pipe-like orifice, and creating a net fluid pressure which varies as the algebraic sum of the pressure in said body and the pressure drop across said orifice.

13. Hydraulic apparatus for producing a pressure variation substantially proportional to the sum of the changes and the time derivative of the changes in a variable condition, comprising, in combination, a fluid chamber, means for varying the pressure in said chamber in accordance with changes in said condition, means associated with said chamber for changing its volume in accordance with changes in the pressure therein, and an orifice having a linear pressure vs. flow characteristic communicating with said chamber and through which fluid flows as a result of changes in said chamber volume, and means for creating a not pressure which varies as the algebraic sum of the pressure in said chamber and the pressure drop across said orifice.

14' Apparatus for producing a net output signal proportional to the sum of the changes and the first time derivative of the changes in a variable condition, the combination comprising a fluid chamber, means for varying the pressure of fluid within said chamber in proportion to changes in said condition, a resilient deformable wall forming a part of said chamber and which is expanded or contracted in proportion to the variations of pressure in said chamber to proportionally change the volume of the latter, a pipe-like orifice communicating with said chamber and through which fluid flows as the volume of the chamber changes, and means for creating a net output signal which varies as the algebraic sum of the pressure in said chamber and the pressure drop across said orifice.

15. Apparatus for creating a hydraulic pressure which varies as the sum of the changes and the time derivative of changes in a variable condition, comprising, in combination, a self-balancing transducer for converting changes in said condition into corresponding changes in liquid pressure within a chamber, resilient means associated with said chamber for changing the volume of the latter in accordance with changes of pressure therein, an orifice having a linear flow vs. pressure characteristic communicating with said chamber and through which fluid flows to or from said transducer as the volume of said chamber changes, and means for creating a net pressure which varies as the algebraic sum of the pressure in said chamber and the pressure drop across said orifice.

16. Hydraulic apparatus comprising, in combination, a self-balancing transducer for converting changes of a variable condition into corresponding changes in a fluid pressure, said transducer having a plunger forming a valve element and a balancing piston respectively movable in a valve chamber and a balancing chamber, said valve chamber being adapted for connection to a source of fluid pressure and a sump and having a port respectively connected to the latter by said element as said plunger is shifted in opposite directions from a neutral position, means for applying a force to said plunger which changes in accordance with changes in said condition from a reference value, an orifice having a linear flow vs. pressure characteristic connected between said port and said balancing chamber, a second chamber connected 115 through saidorifice to said port, resilient means associated with said second chamber for making the volume of the latter change in accordance with the pressure therein, said balancing chamber and piston being sized relative to said second chamber such that the fluid flow into or out of the balancing chamber in response to a change in 16 said condition is negligible relative to the resulting fluid flow into or out of said second chamber, and means for creating a net pressure which varies as the algebraic sum of the pressure in said second chamber and the pressure 5 drop across said orifice.

No references cited. 

